history of mathematics - National STEM Centre

96
Descartes
Figure 7.11 shows a geometrical instrument described in De Orgcmica Conicarum
Sectionum in Piano Descriptione Tractatus (Leiden, 1646) by Frans van Schooten.
fixed ruler E
Figure 7.11
B
B is fixed
to the paper
P U
The ruler E is fixed firmly on the paper. The strips BF, BH, GF and GH are equal in
length. At the point G, which is the 'hinge' of the strips FG, GH and GP, is a pin
that can be moved along a slot in the fixed ruler E. During this movement, the strip
GP remains perpendicular to the ruler E. The 'hinge' B of the strips FB and BH is
fixed to the paper by using a pin. The point H can move freely along the strip FK
(by moving a pin in a slot). Finally, at D, the point of intersection of the strips GP
and FK which can move in a slot in the strip GP, there is a pen which traces out a
curve as G moves along the ruler.
Activity 7.12 Reflecting on Descartes, 17
1 By making a good choice for an x-y coordinate system, show algebraically that
the curve produced by the instrument in Figure 7.11 is part of a parabola.
2 Alternatively, every point of a parabola has the same distance to a fixed point,
called the focus, and a fixed line called the directrix. Prove that the point D has this
property. Use similar triangles to prove that BD = GD.
3 Does this instrument fulfil the conditions that Descartes sets for the movements
of such instruments?
In 1650, Christiaan Huygens (1629-1695), a student of Frans van Schooten, and
thus indirectly of Descartes, made a sketch of a mechanism to construct a spiral.
This mechanism is shown in Figure 7.12.